Vibration responses of nonlinear or time-varying dynamical systems are always nonstationary. Time–frequency representation becomes a necessary approach to analysis such signals. In this paper, a nonstationary vibration analysis method based on continuous wavelet transform (CWT) and Wigner–Ville distribution (WVD) is presented. In order to avoid the cross-terms in the original WVD, a time–frequency filter created by wavelet spectrum is employed to filter the time–frequency distribution (TFD). This process eliminates cross-terms and maintains high time–frequency resolution. The improved WVD is applied to both simulated and practical time-varying systems. Bat echolocation signal, train wheel vibration, and bridge vibration under a moving train are used to assess the proposed method. Comparison results show that the improved WVD is free of cross-terms, effective in identifying time-varying frequencies and is more accurate than the wavelet time–frequency spectrum.

References

References
1.
Ghaffari
,
A.
,
Jafar
,
R.
, and
Morteza
,
T.
,
2007
, “
Time-Varying Transfer Function Extraction of an Unstable Launch Vehicle Via Closed-Loop Identification
,”
Aerosp. Sci. Technol.
,
11
(
2
), pp.
238
244
.
2.
Mourelatos
,
Z. P.
,
Majcher
,
M.
, and
Geroulas
,
V.
,
2016
, “
Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters
,”
ASME J. Vib. Acoust.
,
138
(
3
), p.
031007
.
3.
Mitu
,
A. M.
,
Sireteanu
,
T.
,
Giuclea
,
M.
, and
Solomon
,
O.
,
2016
, “
Simulation of Wide-Sense Stationary Random Time-Series With Specified Spectral Densities
,”
ASME J. Vib. Acoust.
,
138
(
3
), p.
031011
.
4.
Qian
,
S.
, and
Chen
,
D.
,
1999
, “
Joint Time-Frequency Analysis
,”
IEEE Signal Process. Mag.
,
16
(
2
), pp.
52
67
.
5.
Ta
,
M.-N.
, and
Lardiès
,
J.
,
2006
, “
Identification of Weak Nonlinearities on Damping and Stiffness by the Continuous Wavelet Transform
,”
J. Sound Vib.
,
293
(
1
), pp.
16
37
.
6.
Le
,
T.-P.
, and
Paultre
,
P.
,
2012
, “
Modal Identification Based on Continuous Wavelet Transform and Ambient Excitation Tests
,”
J. Sound Vib.
,
331
(
9
), pp.
2023
2037
.
7.
Ruzzene
,
M.
,
Fasana
,
A.
,
Garibaldi
,
L.
, and
Piombo
,
B.
,
1997
, “
Natural Frequencies and Dampings Identification Using Wavelet Transform: Application to Real Data
,”
Mech. Syst. Signal Process.
,
11
(
2
), pp.
207
218
.
8.
Le
,
T.-P.
, and
Argoul
,
P.
,
2004
, “
Continuous Wavelet Transform for Modal Identification Using Free Decay Response
,”
J. Sound Vib.
,
277
(
1
), pp.
73
100
.
9.
Ghanem
,
R.
, and
Romeo
,
F.
,
2000
, “
A Wavelet-Based Approach for the Identification of Linear Time-Varying Dynamical Systems
,”
J. Sound Vib.
,
234
(
4
), pp.
555
576
.
10.
Xu
,
X.
,
Shi
,
Z.
, and
You
,
Q.
,
2012
, “
Identification of Linear Time-Varying Systems Using a Wavelet-Based State-Space Method
,”
Mech. Syst. Signal Process.
,
26
, pp.
91
103
.
11.
Peng
,
Z. K.
, and
Chu
,
F. L.
,
2004
, “
Application of the Wavelet Transform in Machine Condition Monitoring and Fault Diagnostics: A Review With Bibliography
,”
Mech. Syst. Signal Process.
,
18
(
2
), pp.
199
221
.
12.
Staszewski
,
W. J.
, and
Wallace
,
D. M.
,
2014
, “
Wavelet-Based Frequency Response Function for Time-Variant Systems—An Exploratory Study
,”
Mech. Syst. Signal Process.
,
47
(
1
), pp.
35
49
.
13.
Feng
,
Z.
,
Liang
,
M.
, and
Chu
,
F.
,
2013
, “
Recent Advances in Time–Frequency Analysis Methods for Machinery Fault Diagnosis: A Review With Application Examples
,”
Mech. Syst. Signal Process.
,
38
(
1
), pp.
165
205
.
14.
Tang
,
B.
,
Liu
,
W.
, and
Song
,
T.
,
2010
, “
Wind Turbine Fault Diagnosis Based on Morlet Wavelet Transformation and Wigner-Ville Distribution
,”
Renewable Energy
,
35
(
12
), pp.
2862
2866
.
15.
Zhang
,
H. X.
,
Li
,
J.
,
Lin
,
Q.
,
Qu
,
J. Z.
, and
Yang
,
Q. Z.
,
2013
, “
Contrast of Time-Frequency Analysis Methods and Fusion of Wigner-Ville Distribution and Wavelet Transform
,”
Adv. Mater. Res.
,
805
, pp.
1962
1965
.
16.
Shin
,
Y. S.
, and
Jeon
,
J. J.
,
1993
, “
Pseudo Wigner–Ville Time-Frequency Distribution and Its Application to Machinery Condition Monitoring
,”
Shock Vib.
,
1
(
1
), pp.
65
76
.
17.
Lerga
,
J.
, and
Sucic
,
V.
,
2009
, “
Nonlinear IF Estimation Based on the Pseudo WVD Adapted Using the Improved Sliding Pairwise ICI Rule
,”
IEEE Signal Process. Lett.
,
16
(
11
), pp.
953
956
.
18.
Ping
,
D.
,
Liu
,
X.
, and
Deng
,
B.
,
2009
, “
Cross-Term Suppression in the Wigner-Ville Distribution Using Beamforming
,”
Beamforming IEEE Conference on Industrial Electronics and Applications
(
ICIEA
), Xi'an, China, May 25–27, pp.
1103
1105
.
19.
Cai
,
Y. P.
,
Li
,
A. H.
,
Li
,
R. B.
,
Li
,
X. L.
, and
Bai
,
X. F.
,
2011
, “
WVD Cross-Term Suppressing Method Based on Empirical Mode Decomposition
,”
Comput. Eng.
,
37
(
7
), pp.
271
273
.
20.
Mann
,
S.
, and
Haykin
,
S.
,
1992
, “
Adaptive Chirplet Transform: An Adaptive Generalization of the Wavelet Transform
,”
Opt. Eng.
,
31
(
6
), pp.
1243
1256
.
21.
Spanos
,
P.
,
Giaralis
,
A.
, and
Politis
,
N.
,
2007
, “
Time–Frequency Representation of Earthquake Accelerograms and Inelastic Structural Response Records Using the Adaptive Chirplet Decomposition and Empirical Mode Decomposition
,”
Soil Dyn. Earthquake Eng.
,
27
(
7
), pp.
675
689
.
22.
Luo
,
J.
,
Yu
,
D.
, and
Liang
,
M.
,
2012
, “
Gear Fault Detection Under Time-Varying Rotating Speed Via Joint Application of Multiscale Chirplet Path Pursuit and Multiscale Morphology Analysis
,”
Struct. Health Monit.
,
11
(
5
), pp.
526
537
.
23.
Luo
,
J.
,
Yu
,
D.
, and
Liang
,
M.
,
2012
, “
Application of Multi-Scale Chirplet Path Pursuit and Fractional Fourier Transform for Gear Fault Detection in Speed Up and Speed-Down Processes
,”
J. Sound Vib.
,
331
(
22
), pp.
4971
4986
.
24.
Chen
,
G.
,
Chen
,
J.
, and
Dong
,
G.
,
2013
, “
Chirplet Wigner–Ville Distribution for Time–Frequency Representation and Its Application
,”
Mech. Syst. Signal Process.
,
41
(
1
), pp.
1
13
.
25.
Wang
,
M.
,
Chan
,
A. L.
, and
Chui
,
C. K.
,
1996
, “
Wigner-Ville Distribution Decomposition Via Wavelet Packet Transform
,”
IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis
(
TFSA
), Paris, France, June 18–21, pp.
413
416
.
26.
Cohen
,
I.
,
Raz
,
S.
, and
Malah
,
D.
,
1999
, “
Adaptive Suppression of Wigner Interference-Terms Using Shift-Invariant Wavelet Packet Decompositions
,”
Signal Process.
,
73
(
3
), pp.
203
223
.
27.
Hlawatsch
,
F.
, and
Kozek
,
W.
,
1994
, “
Time-Frequency Projection Filters and Time-Frequency Signal Expansions
,”
IEEE Trans. Signal Process.
,
42
(
12
), pp.
3321
3334
.
28.
Feng
,
Z.
, and
Chu
,
F.
,
2007
, “
Nonstationary Vibration Signal Analysis of a Hydroturbine Based on Adaptive Chirplet Decomposition
,”
Struct. Health Monit.
,
6
(
4
), pp.
265
279
.
29.
Baraniuk
,
R.
,
2009
,
Bat Echolocation Chirp
,
DSP Group, Rice University
,
Houston, TX
.
30.
Wang
,
S.
,
Chen
,
X.
,
Wang
,
Y.
,
Cai
,
G.
,
Ding
,
B.
, and
Zhang
,
X.
,
2015
, “
Nonlinear Squeezing Time–Frequency Transform for Weak Signal Detection
,”
Signal Process.
,
113
, pp.
195
210
.
31.
Chen
,
G.
,
Chen
,
J.
,
Dong
,
G.
, and
Jiang
,
H.
,
2015
, “
An Adaptive Non-Parametric Short-Time Fourier Transform: Application to Echolocation
,”
Appl. Acoust.
,
87
, pp.
131
141
.
32.
Liu
,
W.
,
Ma
,
W.
,
Luo
,
S.
,
Zhu
,
S.
, and
Wei
,
C.
,
2015
, “
Research Into the Problem of Wheel Tread Spalling Caused by Wheelset Longitudinal Vibration
,”
Veh. Syst. Dyn.
,
53
(
4
), pp.
546
567
.
33.
Liu
,
W.
,
Zhong
,
L.
,
Luo
,
S.
, and
He
,
X.
,
2016
, “
Research Into the Problem of Polygonal Wheel Wear on the Metro Train
,”
Proc. Inst. Mech. Eng., Part F
,
230
(
1
), pp.
43
55
.
You do not currently have access to this content.